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INPE – National Institute for Space ResearchSão José dos Campos – SP – Brazil – May 16-20, 2016
SIMULATION OF CHUA’S CIRCUIT BY MEANS OF INTERVAL ANALYSIS
Silva, Melanie Rodrigues1, Nepomuceno, E. G.2, Amaral, G. F. V.3, Silva, V. V. R.4
1Modelling and Control Research Group - GCOM, Graduate Program in Electrical Engineering - PPGEL, Federal University of São Joãodel-Rei - UFSJ, São João del-Rei, Brazil, [email protected]
2Modelling and Control Research Group - GCOM, Department of Electrical Engineering - DEPEL, Federal University of São João del-Rei- UFSJ, São João del-Rei, Brazil, [email protected]
3Modelling and Control Research Group - GCOM, Department of Electrical Engineering - DEPEL, Federal University of São João del-Rei- UFSJ, São João del-Rei, Brazil, [email protected]
4Department of Electrical Engineering - DEPEL, Federal University of São João del-Rei - UFSJ, São João del-Rei, Brazil,[email protected]
Abstract: The Chua’s circuit is a paradigm for nonlinear sci-entific studies. It is usually simulated by means of numeri-cal methods under IEEE 754-2008 standard. Although theerror propagation problem is well known, little attention hasbeen given to the relationship between this error and inequal-ities presented in Chua’s circuit model. Taking the average ofround mode towards +∞ and −∞, we showed a qualitativechange on the dynamics of Chua’s circuit.
keywords: Bifurcation Analysis and Applications, Model-ing, Chua’s Circuit, Numerical Simulation and Optimization,Analysis and Control of Nonlinear Dynamical Systems withPractical Applications.
1. INTRODUCTION
We live in a complex world [6] where the majority ofdynamical systems are nonlinear. The choice of non-linearmodels brings with it an inevitable increase in the complex-ity [1]. One of the systems most widely used as a model tostudy the nonlinear dynamics and chaos is the Chua’s circuit[5].
The Chua’s circuit, developed by Leon O. Chua in 1984,is known to exhibit chaotic behaviour similar to the systemproposed by Lorenz [2]. It is a simple electronic networkWhich exhibits a variety of phenomena, such as strange at-tractors and bifurcations. The circuit Consists of two capac-itors, an inductor, a linear resistor, and a nonlinear resistor[5].
The numerical computation uses floating point arithmeticin most computers under the IEEE 7542-2008 standard [3, 4].This standard, in some sense, is a systematic approximationof the real arithmetic and it is represented by a finite subsetof the real numbers. As a result, some properties of the arith-
metic of real numbers are not guaranteed for the floating-point [4]. So, it is necessary attention to the limitations ofcomputers with respect to scientific computing.
Given the arithmetic floating point is an approximationof the real numbers, small errors are generated during theprocess and simulations [8]. These errors in the eyes of manyusers and are made subjectively considered unnoticeable, butcan strongly influence the results.
This article focuses on the computational numerical roundmodes. The inequalities presented the Chua’s circuit modelare addressed in parallel with interval analysis [7] of the cir-cuit. Taking the average of round mode towards +∞ and−∞, we showed a qualitative change on the dynamics ofChua’s circuit.
2. OBJECTIVES
The purpose of this article is to show the influence ofcomputer round modes applied to Chua’s circuit. Using inter-val arithmetic circuit in the simulation mode to a careful anal-ysis of these results and those originating traditional compu-tational rounding, that is, according to the rules of IEEE stan-dard 754-2008 floating-point arithmetic. Since the circuit ischaotic behaviour, the influence of error propagation in thesystem dynamics is also the target of the study.
3. METHODS
The Chua’s circuit is one of the most used systems as anexample to study the non-linear dynamics and chaos. Thecircuit shown in Figure 1a is composed of linear elements,except the Chua’s diode, which shows non-linear behaviouras shown in Figure 1b. This non-linear element can be im-plemented by operational amplifiers [1].The Equations of the
circuit are 1, 2 and 3.
C1dvc1dt
=vc2 − vc1
R− id(vc1) (1)
C2dvc2dt
=vc1 − vc2
R− id (2)
LdiLdt
= −vc2 (3)
(a)
(b)
Figure 1: The Chua’s circuit [1].(a) the Chua’ circuit and (b) curve of Chua diode (current-voltage characteristic).
Since VC1is the voltage across the capacitor C1, VC2
isthe voltage across the capacitor C2, iL is the current throughthe inductor and the current in the diode is[1]:
id(vC1) =
m0vC1+ Bp(m0 −m1) for vC1
< −Bp
m1vC1 for |vC1 | ≤ Bp
m0vC1 + Bp(m1 −m0) for vC1) > Bp
(4)The component values and constants used in the simula-
tions are shown in Table 1. The Algorithms 1 and 2 used inthis work are presented in the Appendix.
Algorithm 1 shows the procedure to decrease the errorpropgation. We use classic RK4, but for each time step, we
Table 1: Values of components and constants used in the sim-ulations.
Components ValuesC1 10 nFC2 100 nFL 19 mHR 1980 Ωm0 -0.37 mSm1 -0.68 mSBp 1.1 V
make two calculations, one using round mode towards −∞and other towards +∞. Algorithm 2 shows that we onlymake the use of average, that is, we get the mid position foreach state variable, from the interval established by the tworound modes. This seems quite naive, but it is not. The stan-dard round mode is the round to nearest. In some way, itworks perfectly when one is interested in just to get the re-sult for only arithmetic operations or to store a real value.But there is no straight way to make a correct round, whenthe most important thing is a function, or a complex set ofoperations. When we make the average, we are in some wayapproaching the round to the nearest, but from the point ofview of functions, what we may call as round to the function.Additionally, as the error coming from rounding off and trun-cation can be se as random (or at least pseudo-random), theaverage can be also seen as a filter to reduce the total error.
4. RESULTS
The results of Chua’s circuit simulation according to thevalues in Table 1 and using the IEEE 754-2008 standard forfloating point are shown in Figures 2a and 3a. In these firstresults we use an traditional approach by means of RK4.However, the results from the solution proposed by Algo-rithm 1, but using the same parameters and initial condition,as shown in Table 1, are presented in Figures 2b and 3b.
It is known that the diode curve, Figure 1b, is given bythe characteristic voltage versus current and in Figure 2 hasthe diode voltage. Thus, it is found by applying the intervalsanalysis Chua’s circuit, the voltage begins to show a periodicbehavior, Figure 2b. Thus, the diode curve changes and thusthe dynamics of the system is affected as shown in Figure 3b.
The orbit of a chaotic dynamic system is sensitive to ini-tial conditions and aperiodic [9]. Although, for the same setof parameters and initial conditions, Figure 2b shows a peri-odical result, which is not sensitivity to initial condition.
5. CONCLUSION
It is observed that even though infinitesimal errors alongthe simulation can compromise the result. Interval analysiswere employed as the round mode was adapted to set themid value of an interval composed by the round modes to-wards to −∞ and +∞. So, what it could be noticed is that
(a)
(b)
Figure 2: Voltage in Chua’s diode.(a) Voltage in the Chua’s diode when the simulation wastraditionally performed using standard IEEE 754-2008. (b)Voltage in the Chua’s diode when the simulation is per-formed by the average of the two round modes towards to−∞ and +∞.
the system simulated by tradition RK4 and IEEE-754-2008round mode standard, that is, round to the nearest, presents astrange attractor with a probably chaotic behaviour. On thecontrary, the proposed algorithm furnishes a beautiful peri-odical result. Which one is the correct? We do not knowfor sure. The simplicity of a periodical solution and the rea-sons presented to use an average of the round methods, helpus to believe that this is the correct solution. We intend tobuild Chua’s circuit following the steps of [5] and comparethis simulated results with experimental results. Although,this procedure seems a way to give a final answer, we shouldkeep in mind that there is no way to implement a circuit thatmatches perfectly the set of equations and parameters givenin this paper. This observation makes our investigation yetmore fascinating!
6. Bibliography
[1] Aguirre, L. A. (2007). Introdução à Identificação deSistemas - Técnicas Lineares e Não-Lineares Aplicadasa Sistemas Reais. Editora da UFMG. 3a edição.
−50
5−1
0
1−4
−2
0
2
4x 10
−3
(a)
−50
5−1
0
1−4
−2
0
2
4x 10
−3
(b)
Figure 3: Scroll of Chua’s circuit.(a) Double attractor traditionally generated. (b) Periodicalresult by the proposed algorithm.
[2] Chua, L. O. (1994). Chua’s circuit: Ten years later. 11.IEEE Trans. Fundamentals., E-77A:1811–1822.
[3] Goldberg, D. (1991). What every computer scientistshould know about floating-point arithmetic. ComputingSurveys, 23:5âAS48.
[4] Institute of Electrical and Electronics Engineers (IEEE)(2008). IEEE 754 standard for floating-point arithmetic.
[5] Kennedy, M. P. (1992). Robust op amp realizaion ofchua’s circuit. Frequenz, 46(3-4):66–80.
[6] Monteiro, L. H. A. (2002). Sistemas Dinâmicos. EditoraLivraria da Física.
[7] Moore, R. E., Bierbaum, F., and Schwiertz, K.-P. (1979).Methods and applications of interval analysis, volume 2.SIAM.
[8] Nepomuceno, E. G. (2014). Convergence of recursivefunctions on computers. The Journal of Engineering,pages 1–3.
[9] S. T. Alligood, T. D. Sauer, J. A. Y. (1997). Chaos: AnIntroduction to Dynamical Systems. Springer.
7. APPENDIX
7.1. Algorithm 1
%Initial Conditionsclear allsystem_dependent('setround',-Inf);ym=[-0.7 0 0];system_dependent('setround',Inf); XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXyp=[-0.7 0 0];tf=0.075;h=1e-6;tspan = 0:h:tf;N=length(tspan);for k=1:N-1
system_dependent('setround',-Inf);aux = ode4(@chua,tspan(k:k+1),ym(k,:),yp(k,:));ym(k+1,:)=aux(2,:);system_dependent('setround',Inf);aux = ode4(@chua,tspan(k:k+1),yp(k,:),ym(k,:));yp(k+1,:)=aux(2,:);
end%Figuresfigure(1)plot(1:N,ym(:,1),1:N,yp(:,1),'k')figure(2)plot3(ym(:,1),ym(:,2),ym(:,3),'k')view(-16,24);grid;
7.2. Algorithm 2
function out = chua(t,in,in2)x = in(1); y = in(2); z = in(3);x2=in2(1);y2=in2(2);z2=in2(3);L = 19.2*10^(-3);C1 = 10*10^(-9);C2 = 100*10^(-9);R = 1978.5;G = 1/R;m0=-0.37*10^(-3);m1=-0.68*10^(-3);Bp=1.1;%Average +Inf and -Infx=(x+x2)/2;y=(y+y2)/2;z=(z+z2)/2;%Diodeif x >Bpg=m0*x+Bp*(m1-m0);
elseif (x >= -Bp)&(x <= Bp)g=m1*x;
elseg=m0*x+Bp*(m0-m1);
end% Chua's Circuit Equationsxdot = (1/C1)*(G*(y-x)-g);ydot = (1/C2)*(G*(x-y)+z);zdot = -(1/L)*y;out = [xdot ydot zdot]';
